Expand knowledge base

6 files changed, 550 insertions, 136 deletions

diff --git a/content/know/concept/einstein-coefficients/index.pdc b/content/know/concept/einstein-coefficients/index.pdc
index bd8f76c..80707c6 100644
--- a/content/know/concept/einstein-coefficients/index.pdc
+++ b/content/know/concept/einstein-coefficients/index.pdc
@@ -136,115 +136,31 @@ This situation is mandatory for lasers, where stimulated emission must dominate,
 such that the light becomes stronger as it travels through the medium.
 
 
-## Electric dipole approximation
+## Coherent light
 
-In fact, we can analytically calculate the Einstein coefficients,
-if we make a mild approximation.
-Consider the Hamiltonian of an electron with charge $q = - e$:
-
-$$\begin{aligned}
-    \hat{H}
-    &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V
-\end{aligned}$$
-
-With $\vec{A}(\vec{r}, t)$ the electromagnetic vector potential.
-We reduce this by fixing the Coulomb gauge $\nabla \!\cdot\! \vec{A} = 0$,
-such that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$,
-and by assuming that $\vec{A}{}^2$ is negligible:
-
-$$\begin{aligned}
-    \hat{H}
-    &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{m} \vec{P} \cdot \vec{A} + V
-\end{aligned}$$
-
-The last term is the Coulomb interaction
-between the electron and the nucleus.
-We can interpret the second term,
-involving the weak $\vec{A}$, as a perturbation $\hat{H}_1$:
-
-$$\begin{aligned}
-    \hat{H}
-    = \hat{H}_0 + \hat{H}_1
-    \qquad \quad
-    \hat{H}_0
-    \equiv \frac{\vec{P}{}^2}{2 m} + V
-    \qquad \quad
-    \hat{H}_1
-    \equiv - \frac{q}{m} \vec{P} \cdot \vec{A}
-\end{aligned}$$
-
-Suppose that $\vec{A}$ is oscillating sinusoidally in time and space as follows:
-
-$$\begin{aligned}
-    \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
-\end{aligned}$$
-
-The corresponding perturbative
-[electric field](/know/concept/electric-field/) $\vec{E}$
-points in the same direction:
-
-$$\begin{aligned}
-    \vec{E}(\vec{r}, t)
-    = - \pdv{\vec{A}}{t}
-    = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
-\end{aligned}$$
-
-Where $\vec{E}_0 = i \omega \vec{A}_0$.
-Let us restrict ourselves to visible light,
-whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$.
-By comparison, the size of an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$,
-so we can ignore the dot product $\vec{k} \cdot \vec{r}$.
-This is the **electric dipole approximation**:
-the radiation is treated classicaly,
-while the electron is treated quantum-mechanically.
-
-$$\begin{aligned}
-    \vec{E}(\vec{r}, t)
-    \approx \vec{E}_0 \exp\!(- i \omega t)
-\end{aligned}$$
-
-Next, we want to convert $\hat{H}_1$
-to use the electric field $\vec{E}$ instead of the potential $\vec{A}$.
-To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$
-and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/):
-
-$$\begin{aligned}
-    \matrixel{2}{\dv*{\vec{r}}{t}}{1}
-    &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1}
-    = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1}
-    \\
-    &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1}
-    = i \omega_0 \matrixel{2}{\vec{r}}{1}
-\end{aligned}$$
-
-Therefore, $\vec{P} / m = i \omega_0 \vec{r}$,
-where $\omega_0 = (E_2 - E_1) / \hbar$ is the resonance frequency of the transition,
-close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating.
-We thus get:
+In fact, we can analytically calculate the Einstein coefficients in some cases,
+by treating incoming light as a perturbation
+to an electron in a two-level system,
+and then finding $B_{12}$ and $B_{21}$ from the resulting transition rate.
+We need to make the [electric dipole approximation](/know/concept/electric-dipole-approximation/),
+in which case the perturbing Hamiltonian $\hat{H}_1(t)$ is given by:
 
 $$\begin{aligned}
     \hat{H}_1(t)
-    &= - \frac{q}{m} \vec{P} \cdot \vec{A}
-    = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
-    \\
-    &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
-    = - \vec{p} \cdot \vec{E}_0 \exp\!(- i \omega t)
+    = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t)
 \end{aligned}$$
 
-Where $\vec{p} \equiv q \vec{r} = - e \vec{r}$ is the electric dipole moment of the electron,
-hence the name *electric dipole approximation*.
-Finally, because electric fields are actually real
-(we made it complex for mathematical convenience),
-we take the real part, yielding:
+Where $q = -e$ is the electron charge,
+$\vec{r}$ is the position operator,
+and $\vec{E}_0$ is the amplitude of
+the [electromagnetic wave](/know/concept/electromagnetic-wave-equation/).
+For simplicity, we let the amplitude be along the $z$-axis:
 
 $$\begin{aligned}
     \hat{H}_1(t)
-    = - q \vec{r} \cdot \vec{E}_0 \cos\!(- i \omega t)
+    = - q E_0 z \cos\!(\omega t)
 \end{aligned}$$
 
-
-## Polarized light
-
 This form of $\hat{H}_1$ is a well-known case for
 [time-dependent perturbation theory](/know/concept/time-dependent-perturbation-theory/),
 which tells us that the transition probability from $\ket{a}$ to $\ket{b}$ is:
@@ -259,19 +175,20 @@ then generally $\ket{1}$ and $\ket{2}$ will be even or odd functions of $z$,
 such that $\matrixel{1}{z}{1} = \matrixel{2}{z}{2} = 0$, leading to:
 
 $$\begin{gathered}
-    \matrixel{1}{H_1}{2} = - q E_0 U
+    \matrixel{1}{H_1}{2} = - E_0 d
     \qquad
-    \matrixel{2}{H_1}{1} = - q E_0 U^*
+    \matrixel{2}{H_1}{1} = - E_0 d^*
     \\
     \matrixel{1}{H_1}{1} = \matrixel{2}{H_1}{2} = 0
 \end{gathered}$$
 
-Where $U \equiv \matrixel{1}{z}{2}$ is a constant.
+Where $d \equiv q \matrixel{1}{z}{2}$ is a constant,
+namely the $z$-component of the **transition dipole moment**.
 The chance of an upward jump (i.e. absorption) is:
 
 $$\begin{aligned}
     P_{12}
-    = \frac{q^2 E_0^2 |U|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+    = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
 \end{aligned}$$
 
 Meanwhile, the transition probability for stimulated emission is as follows,
@@ -280,7 +197,7 @@ and is therefore symmetric around $\omega_{ba}$:
 
 $$\begin{aligned}
     P_{21}
-    = \frac{q^2 E_0^2 |U|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+    = \frac{E_0^2 |d|^2}{\hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
 \end{aligned}$$
 
 Surprisingly, the probabilities of absorption and stimulated emission are the same!
@@ -289,8 +206,12 @@ the availability of electrons and holes in both states.
 
 In theory, we could calculate the transition rate $R_{12} = \pdv*{P_{12}}{t}$,
 which would give us Einstein's absorption coefficient $B_{12}$,
-for this particular case of coherent monochromatic light.
-However, the result would not be constant in time $t$.
+for this specific case of coherent monochromatic light.
+However, the result would not be constant in time $t$,
+so is not really useful.
+
+
+## Polarized light
 
 To solve this "problem", we generalize to (incoherent) polarized polychromatic light.
 To do so, we note that the energy density $u$ of an electric field $E_0$ is given by:
@@ -301,11 +222,12 @@ $$\begin{aligned}
     E_0^2 = \frac{2 u}{\varepsilon_0}
 \end{aligned}$$
 
-Putting this in the previous result gives the following transition probability:
+Where $\varepsilon_0$ is the vacuum permittivity.
+Putting this in the previous result for $P_{12}$ gives us:
 
 $$\begin{aligned}
     P_{12}
-    = \frac{2 u q^2 |U|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
+    = \frac{2 u |d|^2}{\varepsilon_0 \hbar^2} \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2}
 \end{aligned}$$
 
 For a continuous light spectrum,
@@ -313,11 +235,11 @@ this $u$ turns into the spectral energy density $u(\omega)$:
 
 $$\begin{aligned}
     P_{12}
-    = \frac{2 q^2 |U|^2}{\varepsilon_0 \hbar^2}
+    = \frac{2 |d|^2}{\varepsilon_0 \hbar^2}
     \int_0^\infty \frac{\sin^2\!\big( (\omega_0 - \omega) t / 2 \big)}{(\omega_0 - \omega)^2} u(\omega) \dd{\omega}
 \end{aligned}$$
 
-From here, we the derivation is similar to that of
+From here, the derivation is similar to that of
 [Fermi's golden rule](/know/concept/fermis-golden-rule/),
 despite the distinction that we are integrating over frequencies rather than states.
 
@@ -329,8 +251,8 @@ which turns out to be $\pi t$:
 
 $$\begin{aligned}
     P_{12}
-    = \frac{q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x}
-    = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t
+    = \frac{|d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \int_{-\infty}^\infty \frac{\sin^2\!\big(x t \big)}{x^2} \dd{x}
+    = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0) \:t
 \end{aligned}$$
 
 From this, the transition rate $R_{12} = B_{12} u(\omega_0)$
@@ -338,8 +260,8 @@ is then calculated as follows:
 
 $$\begin{aligned}
     R_{12}
-    = \pdv{P_{2 \to 1}}{t}
-    = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2} u(\omega_0)
+    = \pdv{P_{12}}{t}
+    = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2} u(\omega_0)
 \end{aligned}$$
 
 Using the relations from earlier with $g_1 = g_2$,
@@ -348,9 +270,9 @@ for a polarized incoming light spectrum:
 
 $$\begin{aligned}
     \boxed{
-        B_{21} = B_{12} = \frac{\pi q^2 |U|^2}{\varepsilon_0 \hbar^2}
+        B_{21} = B_{12} = \frac{\pi |d|^2}{\varepsilon_0 \hbar^2}
         \qquad
-        A_{21} = \frac{\omega_0^3 q^2 |U|^2}{\pi \varepsilon \hbar c^3}
+        A_{21} = \frac{\omega_0^3 |d|^2}{\pi \varepsilon \hbar c^3}
     }
 \end{aligned}$$
 
@@ -363,31 +285,33 @@ and define the polarization unit vector $\vec{n}$:
 
 $$\begin{aligned}
     \matrixel{1}{\hat{H}_1}{2}
-    = - q \matrixel{1}{\vec{r} \cdot \vec{E}_0}{2}
-    = - q E_0 \matrixel{1}{\vec{r} \cdot \vec{n}}{2}
-    = - q E_0 W
+    = - \vec{d} \cdot \vec{E}_0
+    = - E_0 (\vec{d} \cdot \vec{n})
 \end{aligned}$$
 
-The goal is to obtain the average of $|W|^2$,
-where $W \equiv \matrixel{1}{\vec{r} \cdot \vec{n}}{2}$.
+Where $\vec{d} \equiv q \matrixel{1}{\vec{r}}{2}$ is
+the full **transition dipole moment** vector, which is usually complex.
+
+The goal is to calculate the average of $|\vec{d} \cdot \vec{n}|^2$.
 In [spherical coordinates](/know/concept/spherical-coordinates/),
-we integrate over all possible orientations $\vec{n}$ for fixed $\vec{r}$,
-using that $\vec{r} \cdot \vec{n} = |\vec{r}| \cos\!(\theta)$:
+we integrate over all directions $\vec{n}$ for fixed $\vec{d}$,
+using that $\vec{d} \cdot \vec{n} = |\vec{d}| \cos\!(\theta)$
+with $|\vec{d}| \equiv |d_x|^2 \!+\! |d_y|^2 \!+\! |d_z|^2$:
 
 $$\begin{aligned}
-    \expval{|W|^2}
-    = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\matrixel{1}{\vec{r}}{2}|^2 \cos^2(\theta) \sin\!(\theta) \dd{\varphi} \dd{\theta}
+    \expval{|\vec{d} \cdot \vec{n}|^2}
+    = \frac{1}{4 \pi} \int_0^\pi \int_0^{2 \pi} |\vec{d}|^2 \cos^2(\theta) \sin\!(\theta) \dd{\varphi} \dd{\theta}
 \end{aligned}$$
 
 Where we have divided by $4\pi$ (the surface area of a unit sphere) for normalization,
-and $\theta$ is the polar angle between $\vec{n}$ and $\vec{p}$.
+and $\theta$ is the polar angle between $\vec{n}$ and $\vec{d}$.
 Evaluating the integrals yields:
 
 $$\begin{aligned}
-    \expval{|W|^2}
-    = \frac{2 \pi}{4 \pi} |U|^2 \int_0^\pi \cos^2(\theta) \sin\!(\theta) \dd{\theta}
-    = \frac{|U|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi
-    = \frac{|U|^2}{3}
+    \expval{|\vec{d} \cdot \vec{n}|^2}
+    = \frac{2 \pi}{4 \pi} |\vec{d}|^2 \int_0^\pi \cos^2(\theta) \sin\!(\theta) \dd{\theta}
+    = \frac{|\vec{d}|^2}{2} \Big[ \!-\! \frac{\cos^3(\theta)}{3} \Big]_0^\pi
+    = \frac{|\vec{d}|^2}{3}
 \end{aligned}$$
 
 With this additional constant factor $1/3$,
@@ -395,16 +319,17 @@ the transition rate $R_{12}$ is modified to:
 
 $$\begin{aligned}
     R_{12}
-    = \frac{\pi q^2 |U|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0)
+    = \pdv{P_{12}}{t}
+    = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2} u(\omega_0)
 \end{aligned}$$
 
 From which it follows that the Einstein coefficients for unpolarized light are given by:
 
 $$\begin{aligned}
     \boxed{
-        B_{21} = B_{12} = \frac{\pi q^2 |U|^2}{3 \varepsilon_0 \hbar^2}
+        B_{21} = B_{12} = \frac{\pi |\vec{d}|^2}{3 \varepsilon_0 \hbar^2}
         \qquad
-        A_{21} = \frac{\omega_0^3 q^2 |U|^2}{3 \pi \varepsilon \hbar c^3}
+        A_{21} = \frac{\omega_0^3 |\vec{d}|^2}{3 \pi \varepsilon \hbar c^3}
     }
 \end{aligned}$$
 
diff --git a/content/know/concept/electric-dipole-approximation/index.pdc b/content/know/concept/electric-dipole-approximation/index.pdc
new file mode 100644
index 0000000..96b4fed
--- /dev/null
+++ b/content/know/concept/electric-dipole-approximation/index.pdc
@@ -0,0 +1,147 @@
+---
+title: "Electric dipole approximation"
+firstLetter: "E"
+publishDate: 2021-09-14
+categories:
+- Physics
+- Quantum mechanics
+- Optics
+
+date: 2021-09-14T13:11:54+02:00
+draft: false
+markup: pandoc
+---
+
+# Electric dipole approximation
+
+Suppose that an [electromagnetic wave](/know/concept/electromagnetic-wave-equation/)
+is travelling through an atom, and affecting the electrons.
+The general Hamiltonian of an electron in such a wave is given by:
+
+$$\begin{aligned}
+    \hat{H}
+    &= \frac{\vec{P}{}^2}{2 m} - \frac{q}{2 m} (\vec{A} \cdot \vec{P} + \vec{P} \cdot \vec{A}) + \frac{q^2 \vec{A}{}^2}{2m} + V
+\end{aligned}$$
+
+With charge $q = - e$
+and electromagnetic vector potential $\vec{A}(\vec{r}, t)$.
+We reduce this by fixing the Coulomb gauge $\nabla \cdot \vec{A} = 0$,
+so that $\vec{A} \cdot \vec{P} = \vec{P} \cdot \vec{A}$,
+and assume that $\vec{A}{}^2$ is negligible:
+
+$$\begin{aligned}
+    \hat{H}
+    = \hat{H}_0 + \hat{H}_1
+    \qquad \quad
+    \hat{H}_0
+    \equiv \frac{\vec{P}{}^2}{2 m} + V
+    \qquad \quad
+    \hat{H}_1
+    \equiv - \frac{q}{m} \vec{P} \cdot \vec{A}
+\end{aligned}$$
+
+We have split $\hat{H}$ into $\hat{H}_0$
+and a perturbation $\hat{H}_1$, since $\vec{A}$ is small.
+In an electromagnetic wave,
+$\vec{A}$ is oscillating sinusoidally in time and space as follows:
+
+$$\begin{aligned}
+    \vec{A}(\vec{r}, t) = \vec{A}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+\end{aligned}$$
+
+The corresponding perturbative
+[electric field](/know/concept/electric-field/) $\vec{E}$
+points in the same direction:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = - \pdv{\vec{A}}{t}
+    = \vec{E}_0 \exp\!(i \vec{k} \cdot \vec{r} - i \omega t)
+\end{aligned}$$
+
+Where $\vec{E}_0 = i \omega \vec{A}_0$.
+Let us restrict ourselves to visible light,
+whose wavelength $2 \pi / k \approx 10^{-6} \:\mathrm{m}$.
+Meanwhile, an atomic orbital is on the order of $10^{-10} \:\mathrm{m}$,
+so $\vec{k} \cdot \vec{r}$ is negligible:
+
+$$\begin{aligned}
+    \boxed{
+        \vec{E}(\vec{r}, t)
+        \approx \vec{E}_0 \exp\!(- i \omega t)
+    }
+\end{aligned}$$
+
+This is the **electric dipole approximation**:
+we ignore all spatial variation of $\vec{E}$,
+and only consider its temporal oscillation.
+Also, since we have not used the word "photon",
+we are implicitly treating the radiation classically,
+and the electron quantum-mechanically.
+
+Next, we want to convert $\hat{H}_1$
+to use the electric field $\vec{E}$ instead of the potential $\vec{A}$.
+To do so, we rewrite the momemtum $\vec{P} = m \: \dv*{\vec{r}}{t}$
+and evaluate this in the [Heisenberg picture](/know/concept/heisenberg-picture/):
+
+$$\begin{aligned}
+    \matrixel{2}{\dv*{\vec{r}}{t}}{1}
+    &= \frac{i}{\hbar} \matrixel{2}{[\hat{H}_0, \vec{r}]}{1}
+    = \frac{i}{\hbar} \matrixel{2}{\hat{H}_0 \vec{r} - \vec{r} \hat{H}_0}{1}
+    \\
+    &= \frac{i}{\hbar} (E_2 - E_1) \matrixel{2}{\vec{r}}{1}
+    = i \omega_0 \matrixel{2}{\vec{r}}{1}
+\end{aligned}$$
+
+Therefore, $\vec{P} / m = i \omega_0 \vec{r}$,
+where $\omega_0 \equiv (E_2 - E_1) / \hbar$ is the resonance frequency of the transition,
+close to which we assume that $\vec{A}$ and $\vec{E}$ are oscillating.
+We thus get:
+
+$$\begin{aligned}
+    \hat{H}_1(t)
+    &= - \frac{q}{m} \vec{P} \cdot \vec{A}
+    = - i q \omega_0 \vec{r} \cdot \vec{A}_0 \exp\!(- i \omega t)
+    \\
+    &= - q \vec{r} \cdot \vec{E}_0 \exp\!(- i \omega t)
+    = - \vec{d} \cdot \vec{E}_0 \exp\!(- i \omega t)
+\end{aligned}$$
+
+Where $\vec{d} \equiv q \vec{r} = - e \vec{r}$ is
+the **transition dipole moment operator** of the electron,
+hence the name **electric dipole approximation**.
+Finally, since electric fields are actually real
+(we let it be complex for mathematical convenience),
+we take the real part, yielding:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{H}_1(t)
+        = - q \vec{r} \cdot \vec{E}_0 \cos\!(\omega t)
+    }
+\end{aligned}$$
+
+If this approximation is too rough,
+$\vec{E}$ can always be Taylor-expanded in $(i \vec{k} \cdot \vec{r})$:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = \vec{E}_0 \Big( 1 + (i \vec{k} \cdot \vec{r}) + \frac{1}{2} (i \vec{k} \cdot \vec{r})^2 + \: ... \Big) \exp\!(- i \omega t)
+\end{aligned}$$
+
+Taking the real part then yields the following series of higher-order correction terms:
+
+$$\begin{aligned}
+    \vec{E}(\vec{r}, t)
+    = \vec{E}_0 \Big( \cos\!(\omega t) + (\vec{k} \cdot \vec{r}) \sin\!(\omega t) - \frac{1}{2} (\vec{k} \cdot \vec{r})^2 \cos\!(\omega t) + \: ... \Big)
+\end{aligned}$$
+
+
+
+## References
+1.  M. Fox,
+    *Optical properties of solids*, 2nd edition,
+    Oxford.
+2.  D.J. Griffiths, D.F. Schroeter,
+    *Introduction to quantum mechanics*, 3rd edition,
+    Cambridge.
diff --git a/content/know/concept/electromagnetic-wave-equation/index.pdc b/content/know/concept/electromagnetic-wave-equation/index.pdc
index 68fe062..84946bb 100644
--- a/content/know/concept/electromagnetic-wave-equation/index.pdc
+++ b/content/know/concept/electromagnetic-wave-equation/index.pdc
@@ -118,14 +118,18 @@ $$\begin{aligned}
     \vb{E}(\vb{r}, t)
     &= \vb{E}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
     \\
-    \vb{H}(\vb{r}, t)
-    &= \vb{H}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
+    \vb{B}(\vb{r}, t)
+    &= \vb{B}_0 \exp\!(i \vb{k} \cdot \vb{r} - i \omega t)
 \end{aligned}$$
 
-In fact, thanks to linearity, these solutions can be treated as
+In fact, thanks to linearity, these **plane waves** can be treated as
 terms in a Fourier series, meaning that virtually
 *any* function $f(\vb{k} \cdot \vb{r} - \omega t)$ is a valid solution.
 
+Keep in mind that in reality, $\vb{E}$ and $\vb{B}$ are real,
+so although it is mathematically convenient to use plane waves,
+in the end you will need to take the real part.
+
 
 ## Non-uniform medium
 
diff --git a/content/know/concept/interaction-picture/index.pdc b/content/know/concept/interaction-picture/index.pdc
new file mode 100644
index 0000000..1ce330d
--- /dev/null
+++ b/content/know/concept/interaction-picture/index.pdc
@@ -0,0 +1,214 @@
+---
+title: "Interaction picture"
+firstLetter: "I"
+publishDate: 2021-09-13
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-09-09T21:15:37+02:00
+draft: false
+markup: pandoc
+---
+
+# Interaction picture
+
+The **interaction picture** or **Dirac picture**
+is an alternative formulation of quantum mechanics,
+equivalent to both the Schrödinger picture
+and the [Heisenberg picture](/know/concept/heisenberg-picture/).
+
+Recall that Schrödinger lets states $\ket{\psi_S(t)}$ evolve in time,
+but keeps operators $\hat{L}_S$ fixed (except for explicit time dependence).
+Meanwhile, Heisenberg keeps states $\ket{\psi_H}$ fixed,
+and puts all time dependence on the operators $\hat{L}_H(t)$.
+
+However, in the interaction picture,
+both the states $\ket{\psi_I(t)}$ and the operators $\hat{L}_I(t)$
+evolve in $t$.
+This might seem unnecessarily complicated,
+but it turns out be convenient when considering
+a time-dependent "perturbation" $\hat{H}_{1,S}$
+to a time-independent Hamiltonian $\hat{H}_{0,S}$:
+
+$$\begin{aligned}
+    \hat{H}_S(t)
+    = \hat{H}_{0,S} + \hat{H}_{1,S}(t)
+\end{aligned}$$
+
+With $\hat{H}_S(t)$ the full Schrödinger Hamiltonian.
+We define the unitary conversion operator:
+
+$$\begin{aligned}
+    \hat{U}(t)
+    \equiv \exp\!\Big( i \frac{\hat{H}_{0,S} t}{\hbar} \Big)
+\end{aligned}$$
+
+The interaction-picture states $\ket{\psi_I(t)}$ and operators $\hat{L}_I(t)$
+are then defined to be:
+
+$$\begin{aligned}
+    \boxed{
+        \ket{\psi_I(t)}
+        \equiv \hat{U}(t) \ket{\psi_S(t)}
+        \qquad
+        \hat{L}_I(t)
+        \equiv \hat{U}(t) \: \hat{L}_S(t) \: \hat{U}{}^\dagger(t)
+    }
+\end{aligned}$$
+
+
+## Equations of motion
+
+To find the equation of motion for $\ket{\psi_I(t)}$,
+we differentiate it and multiply by $i \hbar$:
+
+$$\begin{aligned}
+    i \hbar \dv{t} \ket{\psi_I}
+    &= i \hbar \Big( \dv{\hat{U}}{t} \ket{\psi_S} + \hat{U} \dv{t} \ket{\psi_S} \Big)
+    \\
+    &= i \hbar \Big( i \frac{\hat{H}_{0,S}}{\hbar} \Big) \hat{U} \ket{\psi_S} + \hat{U} \Big( i \hbar \dv{t} \ket{\psi_S} \Big)
+\end{aligned}$$
+
+We insert the Schrödinger equation into the second term,
+and use $\comm*{\hat{U}}{\hat{H}_{0,S}} = 0$:
+
+$$\begin{aligned}
+    i \hbar \dv{t} \ket{\psi_I}
+    &= - \hat{H}_{0,S} \hat{U} \ket{\psi_S} + \hat{U} \hat{H}_S \ket{\psi_S}
+    \\
+    &= \hat{U} \big( \!-\! \hat{H}_{0,S} + \hat{H}_S \big) \ket{\psi_S}
+    \\
+    &= \hat{U} \big( \hat{H}_{1,S} \big) \hat{U}{}^\dagger \hat{U} \ket{\psi_S}
+\end{aligned}$$
+
+Which leads to an analogue of the Schrödinger equation,
+with $\hat{H}_{1,I} = \hat{U} \hat{H}_{1,S} \hat{U}{}^\dagger$:
+
+$$\begin{aligned}
+    \boxed{
+        i \hbar \dv{t} \ket{\psi_I(t)}
+        = \hat{H}_{1,I}(t)  \ket{\psi_I(t)}
+    }
+\end{aligned}$$
+
+Next, we do the same with an operator $\hat{L}_I$
+to find a description of its evolution in time:
+
+$$\begin{aligned}
+    \dv{t} \hat{L}_I
+    &= \dv{\hat{U}}{t} \hat{L}_S \hat{U}{}^\dagger + \hat{U} \hat{L}_S \dv{\hat{U}{}^\dagger}{t} + \hat{U} \dv{\hat{L}_S}{t} \hat{U}{}^\dagger
+    \\
+    &= \frac{i}{\hbar} \hat{U} \hat{H}_{0,S} \big( \hat{U}{}^\dagger \hat{U} \big) \hat{L}_S \hat{U}{}^\dagger
+    - \frac{i}{\hbar} \hat{U} \hat{L}_S \big( \hat{U}{}^\dagger \hat{U} \big) \hat{H}_{0,S} \hat{U}{}^\dagger
+    + \Big( \dv{\hat{L}_S}{t} \Big)_I
+    \\
+    &= \frac{i}{\hbar} \hat{H}_{0,I} \hat{L}_I
+    - \frac{i}{\hbar} \hat{L}_I \hat{H}_{0,I}
+    + \Big( \dv{\hat{L}_S}{t} \Big)_I
+    = \frac{i}{\hbar} \comm*{\hat{H}_{0,I}}{\hat{L}_I} + \Big( \dv{\hat{L}_S}{t} \Big)_I
+\end{aligned}$$
+
+The result is analogous to the equation of motion in the Heisenberg picture:
+
+$$\begin{aligned}
+    \boxed{
+        \dv{t} \hat{L}_I(t)
+        = \frac{i}{\hbar} \comm*{\hat{H}_{0,I}(t)}{\hat{L}_I(t)} + \Big( \dv{t} \hat{L}_S(t) \Big)_I
+    }
+\end{aligned}$$
+
+
+## Time evolution operator
+
+Recall that an alternative form of the Schrödinger equation is as follows,
+where a **time evolution operator**  or
+**generator of translations in time** $K_S(t, t_0)$
+brings $\ket{\psi_S}$ from time $t_0$ to $t$:
+
+$$\begin{aligned}
+    \ket{\psi_S(t)}
+    = \hat{K}_S(t, t_0) \ket{\psi_S(t_0)}
+    \qquad \quad
+    \hat{K}_S(t, t_0)
+    \equiv \exp\!\Big( \!-\! i \frac{\hat{H}_S (t - t_0)}{\hbar} \Big)
+\end{aligned}$$
+
+We want to find an analogous operator in the interaction picture, satisfying:
+
+$$\begin{aligned}
+    \ket{\psi_I(t)}
+    \equiv \hat{K}_I(t, t_0) \ket{\psi_I(t_0)}
+\end{aligned}$$
+
+Inserting this definition into the equation of motion for $\ket{\psi_I}$ yields
+an equation for $\hat{K}_I$, with the logical boundary condition $\hat{K}_I(t_0, t_0) = 1$:
+
+$$\begin{aligned}
+    i \hbar \dv{t} \Big( \hat{K}_I(t, t_0) \ket{\psi_I(t_0)} \Big)
+    &= \hat{H}_{1,I}(t) \Big( \hat{K}_I(t, t_0) \ket{\psi_I(t_0)} \Big)
+    \\
+    i \hbar \dv{t} \hat{K}_I(t, t_0)
+    &= \hat{H}_{1,I}(t) \hat{K}_I(t, t_0)
+\end{aligned}$$
+
+We turn this into an integral equation
+by integrating both sides from $t_0$ to $t$:
+
+$$\begin{aligned}
+    i \hbar \int_{t_0}^t \dv{t'} K_I(t', t_0) \dd{t'}
+    = \int_{t_0}^t \hat{H}_{1,I}(t') \hat{K}_I(t', t_0) \dd{t'}
+\end{aligned}$$
+
+After evaluating the left integral,
+we see an expression for $\hat{K}_I$ as a function of $\hat{K}_I$ itself:
+
+$$\begin{aligned}
+     K_I(t, t_0)
+    = 1 + \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \hat{K}_I(t', t_0) \dd{t'}
+\end{aligned}$$
+
+By recursively inserting $\hat{K}_I$ once, we get a longer expression,
+still with $\hat{K}_I$ on both sides:
+
+$$\begin{aligned}
+     K_I(t, t_0)
+    = 1 + \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'}
+    + \frac{1}{(i \hbar)^2} \int_{t_0}^t \hat{H}_{1,I}(t') \int_{t_0}^{t'} \hat{H}_{1,I}(t'') \hat{K}_I(t'', t_0) \dd{t''} \dd{t'}
+\end{aligned}$$
+
+And so on. Note the ordering of the integrals and integrands:
+upon closer inspection, we see that the $n$th term is
+a [time-ordered product](/know/concept/time-ordered-product/) $\mathcal{T}$
+of $n$ factors $\hat{H}_{1,I}$:
+
+$$\begin{aligned}
+    \hat{K}_I(t, t_0)
+    &= 1 + \int_{t_0}^t \hat{H}_{1,I}(t_1) \dd{t_1}
+    + \frac{1}{2} \int_{t_0}^{t} \int_{t_0}^{t_1} \mathcal{T} \Big\{ \hat{H}_{1,I}(t_1) \hat{H}_{1,I}(t_2) \Big\} \dd{t_1} \dd{t_2}
+    + \: ...
+    \\
+    &= 1 + \sum_{n = 1}^\infty \frac{1}{n!} \frac{1}{(i \hbar)^n}
+    \int_{t_0}^{t} \cdots \int_{t_0}^{t_n} \mathcal{T} \Big\{ \hat{H}_{1,I}(t_1) \cdots \hat{H}_{1,I}(t_n) \Big\} \dd{t_1} \cdots \dd{t_n}
+    \\
+    &= \sum_{n = 0}^\infty \frac{1}{n!} \frac{1}{(i \hbar)^n}
+    \mathcal{T} \bigg\{ \bigg( \int_{t_0}^{t} \hat{H}_{1,I}(t') \dd{t'} \bigg)^n \bigg\}
+\end{aligned}$$
+
+Here, we recognize the Taylor expansion of $\exp$,
+leading us to a final expression for $\hat{K}_I$:
+
+$$\begin{aligned}
+    \boxed{
+        \hat{K}_I(t, t_0)
+        = \mathcal{T} \bigg\{ \exp\!\bigg( \frac{1}{i \hbar} \int_{t_0}^t \hat{H}_{1,I}(t') \dd{t'} \bigg) \bigg\}
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.
+
diff --git a/content/know/concept/lorentz-force/index.pdc b/content/know/concept/lorentz-force/index.pdc
index f0e9850..2362766 100644
--- a/content/know/concept/lorentz-force/index.pdc
+++ b/content/know/concept/lorentz-force/index.pdc
@@ -231,7 +231,7 @@ Curiously, $\vb{v}_d$ is independent of $q$.
 Such a drift is not specific to an electric field.
 In the equations above, $\vb{E}$ can be replaced
 by a general force $\vb{F}/q$ (e.g. gravity) without issues.
-In that case, $\vb{v}_d$ does depend on $q$.
+In that case, $\vb{v}_d$ does depend on $q$:
 
 $$\begin{aligned}
     \boxed{
diff --git a/content/know/concept/time-ordered-product/index.pdc b/content/know/concept/time-ordered-product/index.pdc
new file mode 100644
index 0000000..82c9d0f
--- /dev/null
+++ b/content/know/concept/time-ordered-product/index.pdc
@@ -0,0 +1,124 @@
+---
+title: "Time-ordered product"
+firstLetter: "T"
+publishDate: 2021-09-13
+categories:
+- Physics
+- Quantum mechanics
+
+date: 2021-09-13T19:58:33+02:00
+draft: false
+markup: pandoc
+---
+
+# Time-ordered product
+
+In quantum mechanics, especially quantum field theory,
+a **time-ordered product** is a product of
+explicitly time-dependent operators,
+subject to certain ordering constraints.
+
+Let us start with an unusual motivation.
+Suppose that some time-dependent operator $\hat{A}(t)$ is defined like so,
+as a product of $N$ time-dependent sub-operators $\hat{a}_n(t)$:
+
+$$\begin{aligned}
+    \hat{A}(t)
+    \equiv \int_0^{t} \hat{a}_1(t_1) \bigg( \int_0^{t_1} \hat{a}_2(t_2) \bigg( \int_0^{t_2} \hat{a}_3(t_3) \bigg( \cdots \bigg)
+    \dd{t_3} \bigg) \dd{t_2} \bigg) \dd{t_1}
+\end{aligned}$$
+
+Crucially, the upper limits of the inner integrals
+depend on the surrounding variables,
+meaning that these integrals cannot simply be reordered.
+
+An interpretation is that the rightmost $\hat{a}_N(t_N)$ is applied first,
+and then $\hat{a}_{N-1}(t_{N-1})$ secondly with $t_{N-1} > t_N$,
+and so on.
+This suggests there is a form of "time-ordering" here:
+the integrals sweep across all relative timings of $\hat{a}_n$,
+but preserve the ordering.
+Indeed, this could be rewritten as a time-ordered product
+(see the [interaction picture](/know/concept/interaction-picture/) for an example).
+
+A more general and intuitive motivation goes as follows.
+Suppose we have a product of $N$ time-dependent operators $\hat{a}_n(t)$,
+each representing a certain event.
+Clearly, we would want to apply them in chronological order:
+
+$$\begin{aligned}
+    \hat{a}_N(t_N) \: \hat{a}_{N-1}(t_{N-1}) \: \cdots \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+    \qquad \mathrm{where} \qquad
+    t_N > t_{N-1} > ... > \: t_2 > t_1
+\end{aligned}$$
+
+But what if the ordering of the arguments $t_N, ..., t_1$
+is not known in advance?
+We thus define the **time-ordering meta-operator** $\mathcal{T}$,
+which reorders the operators based on the $t$-values
+such that they are always in chronological order.
+For example:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    \equiv
+    \begin{cases}
+        \hat{a}_1(t_1) \: \hat{a}_2(t_2) & \mathrm{if} \; t_2 < t_1 \\
+        \hat{a}_2(t_2) \: \hat{a}_1(t_1) & \mathrm{if} \; t_1 < t_2
+    \end{cases}
+\end{aligned}$$
+
+This example suggests a general algorithm for $\mathcal{T}$:
+we need to consider every permutation of the operators $\hat{a}_n(t_n)$,
+and leave only the single one that satisfies our demands.
+
+Mathematically, we do this by summing up all permutations,
+and multiplying each term with a product of
+[Heaviside step functions](/know/concept/heaviside-step-function/) $\Theta$,
+which remove the term if the ordering is wrong:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\}
+    \equiv \sum_{p \in P_N}^{}
+    \Theta\big(t_{p_1} \!\!-\! t_{p_2}\big) \cdots \Theta\big(t_{p_{N-1}} \!\!-\! t_{p_N}\big)
+    \: \hat{a}_{p_1}(t_{p_1}) \: \cdots \: \hat{a}_{p_N}(t_{p_N})
+\end{aligned}$$
+
+With this, our earlier example for two operators $\hat{a}_1$ and $\hat{a}_2$
+takes the following form:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) + \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+\end{aligned}$$
+
+However, we are still missing an important detail:
+so far, we have quietly been assuming that the operators are bosonic
+(see [second quantization](/know/concept/second-quantization/)).
+To include fermionic operators,
+we must allow the sign of each term to change,
+based on whether the permutation is even or odd:
+
+$$\begin{aligned}
+    \mathcal{T} \big\{ \hat{a}_1(t_1) \: \hat{a}_2(t_2) \big\}
+    = \Theta(t_1 - t_2) \: \hat{a}_1(t_1) \: \hat{a}_2(t_2) \pm \Theta(t_2 - t_1) \: \hat{a}_2(t_2) \: \hat{a}_1(t_1)
+\end{aligned}$$
+
+Where $\pm$ is $+$ for bosons, and $-$ for fermions in this case.
+The general definition of $\mathcal{T}$ is:
+
+$$\begin{aligned}
+    \boxed{
+        \mathcal{T} \big\{ \hat{a}_1 \cdots \hat{a}_N \big\}
+        \equiv \sum_{p \in P_N}^{} (-1)^p
+        \bigg( \prod_{j = 1}^{N-1} \Theta\big(t_{p_j} \!-\! t_{p_{j+1}}\big) \bigg)
+        \bigg( \prod_{k = 1}^N \hat{a}_{p_k}(t_{p_k}) \bigg)
+    }
+\end{aligned}$$
+
+
+
+## References
+1.  H. Bruus, K. Flensberg,
+    *Many-body quantum theory in condensed matter physics*,
+    2016, Oxford.